Suppose that $\mathbb{C}$ is a category with finite limits and that $\mathcal{D}$ is a category internal to $\mathbb{C}$. We can also represent $\mathcal{D}$ as a fibration $\mathbb{D}\to\mathbb{C}$.

My main question is this: What is the difference between an internal presheaf on $\mathcal{D}$ and an ordinary (external) presheaf on $\mathbb{D}$?

Other related questions (or the same question phrased differently):

If $F$ is an internal presheaf on $\mathcal{D}$, we should be able to represent it as an internal fibration $\mathcal{F}\to\mathcal{D}$; how does this differ from an ordinary fibration over $\mathbb{D}$.

Given an internal presheaf, (I think) we should be able to externalize the corresponding internal fibration $\mathcal{F}\to\mathcal{D}$ to an external fibration $\mathbb{F}\to\mathbb{D}$. Is that correct? If so, what further properties does $\mathbb{F}$ satisfy, in addition to being an (external) fibration?

We can raise the same questions for internal sheaves and stacks (say for the regular topology). Do any additional subtleties arise in that situation?

In a brief --- usually internal structures correspond to fibrewise external structures, and (also) in many cases to fibred total structures. Actually, this is true for internal presheaves as well as for internal fibrations. The extra properites needed to state the equivalence between internal and external total structures are induced by the fact that the total categories obtained from the process of externalization are themselves fibred over the base category. If you need more details I may write a full answer after the weekend.
–
Michal R. PrzybylekJun 27 '13 at 17:45

1 Answer
1

Fix a category $\mathbb{C}$ with finite limits. Denote by $\mathbf{Cat}(\mathbb{C})$ the 2-category of $\mathbb{C}$ internal categories. It is well-known that $\mathbf{Cat}(\mathbb{C})$ is a full 2-subcategory of the 2-category $\mathbf{Cat}^{\mathbb{C}^{op}}$ of $\mathbf{Cat}$-valued presheaves on $\mathbb{C}$ via the usual externalization functor:
$$\mathit{fam}(-) \colon \mathbf{Cat}(\mathbb{C}) \rightarrow \mathbf{Cat}^{\mathbb{C}^{op}}$$
By so-called fibred Yoneda lemma (a variant of Yoneda lemma for weak 2-categories) $\mathbf{Cat}^{\mathbb{C}^{op}}$ is weakly 2-equivalent to the category $\mathit{PsFn}(\mathbb{C}^{op}, \mathbf{Cat})$ of pseudofunctors $\mathbb{C}^{op} \rightarrow \mathbf{Cat}$, also called $\mathbb{C}$-indexed categories or variations on $\mathbb{C}$.

Moreover, under assumption of AC in our meta-theory, the Grothendieck construction and its inverse induce equivalence between $\mathbb{C}$-indexed categories and fibrations over $\mathbb{C}$. Fibrations that actually correspond to $\mathbf{Cat}$-valued presheaves are called split; similarly a split cartesian functors between split fibration is a cartesian functor that corresponds to a strict natural transformation.

This means that under sufficiently strong meta-foundations one may equivalently speak of:

These conditions are called "smallness" (which may by characterised as "having generic object and generic morphisms").

Now, if we agree that the above views are equivalent, we may check how internally defined concepts transport along the equivalences. I shall start with fibrations internal to $\mathbf{Cat}(\mathbb{C})$.

Because the concept of an internal fibration is universally characterisable by (weighted-)limits (actually comma objects) and adjunctions, and since the externalization functor $\mathit{fam}(-) \colon \mathbf{Cat}(\mathbb{C}) \rightarrow \mathbf{Cat}^{\mathbb{C}^{op}}$ preserves limits and adjunctions (every 2-functors preserves adjunctions, since adjunctions are equationally defined), a $\mathbf{Cat}(\mathbb{C})$-internal fibration is mapped to an internal fibration between $\mathbf{Cat}$-valued (small) presheaves on $\mathbb{C}$. Therefore, it suffices to characterise fibrations in $\mathbf{Cat}^{\mathbb{C}^{op}}$, or by the above equivalences (to directly address your question), it suffices to characterise fibrations internal to fibrations over $\mathbb{C}$.

So let us assume that there are two fibrations $p, q$ together with a cartesian functor $G$ (that is, a functor that maps cartesian arrows from one fibration to cartesian arrows in the other fibration) over $\mathbb{C}$:
$$\require{AMScd}
\begin{CD}
\mathbb{A} @>{G}>> \mathbb{B}\\
@VpVV @VVqV \\
\mathbb{C} @= \mathbb{C}
\end{CD}$$
Then $G$ is a fibration from $p$ to $q$ internal to fibrations over $\mathbb{C}$ iff $G$ is a fibration in the usual sense between categories $\mathbb{A}$ and $\mathbb{B}$. In terms of "fibrewise" structure it means that for each object $C \in \mathbb{C}$ the fibre $p_C$ of $p$ over $C$ is fibred (in our case: by the restriction of $G$) over the fibre $q_C$ of $q$ over $C$ and the Beck-Chevalley condition (the stability condition) holds.

Now, as you said, an internal presheaf $F \colon \mathcal{D}^{op} \rightarrow \mathbb{C}$ over $\mathcal{D}$ corresponds to a discrete internal fibration $\pi_F \colon \int F \rightarrow \mathcal{D}$. Its externalised version:
$$\require{AMScd}
\begin{CD}
\mathit{Fam}(\int F) @>{\mathit{fam}(\pi_F)}>> \mathit{Fam}(\mathcal{D})\\
@V{\mathit{fam}(\int F)}VV @VV{\mathit{fam}(\mathcal{D})}V \\
\mathbb{C} @= \mathbb{C}
\end{CD}$$
consists of a fibred collection of discrete fibrations (these fibrations are discrete almost by definition of an internal discrete fibration; in fact, the converse is also true). In other words, for each $C \in \mathbb{C}$ the restriction $\mathit{fam}(\pi_F)_C$ of $\mathit{fam}(\pi_F)$ to $C$ is a discrete fibration from fibre $\mathit{fam}(\int F)_C$ to fibre $\mathit{fam}(\mathcal{D})_C$, and so corresponds to a $\mathbf{Set}$-valued preseaf over fibre $\mathit{fam}(\mathcal{D})_C$.

Of course $\mathit{fam}(\pi_F) \colon \mathit{Fam}(\int F) \rightarrow \mathit{Fam}(\mathcal{D})$ thought of as an ordinary fibration between categories is not (generally) discrete, therefore cannot be thought of as a single $\mathbf{Set}$-valued presheaf (by Yoneda the "best $\mathbf{Set}$-valued presheaf approximation" to $\mathit{fam}(\pi_F)$ is the presheaf associated to the discrete fibration $\mathit{fam}(\pi_F)_1$).